Cuboids are canonically Ramsey

Abstract

We say a set of points $C\subset \mathbb{R}^n$ is canonically Ramsey if there is some set of points $S\subset \mathbb{R}^{n'}$ such that any colouring of $S$, with any number of colours, admits either a monochromatic or rainbow copy of $C$ -- that is to say, some set of points congruent to $C$ either all receive the same colour, or all receive different colours. Mao, Ozeki, and Wang introduced this notion, proving that 30-60-90 triangles are canonically Ramsey, since when various other canonically Ramsey configurations have been identified (by Gehér, Sagdeev, and Tóth, and others). Fang, Ge, Shu, Xu, Xu, and Yang showed that all triangles and rectangles are canonically Ramsey, and asked whether all cuboids are canonically Ramsey. Here cuboids are sets of the form $\{0,b_1\}\times\dots\times\{0,b_s\}$, and in particular may have dimension greater than three. We resolve this question, proving that all cuboids are canonically Ramsey.

Theorems & Definitions (8)

• Theorem 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Lemma \ref{['lem:IR']}
• proof : Proof of Theorem \ref{['thm:main']}